Finite Group Databasebeta

← #14788534

Subgroups — #14788534

Order 2816.

1214 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623116C211yesmaximal, characteristic, complemented
2C2×#10622628C211yesmaximal, characteristic
3C2×C4×#4324C211yesmaximal, characteristic, complemented
4#13069857C212yesmaximal, complemented
5#13069692C212yesmaximal, complemented
6#13069553C211yesmaximal, characteristic, complemented
7#13069421C211yesmaximal, characteristic, complemented
8#13068967C212yesmaximal, complemented
9#13068876C212yesmaximal, complemented
10#13068618C211yesmaximal, characteristic, complemented
11C11×#2538C211yesmaximal, characteristic, complemented
12C2×C11×C22⋊Q8—21no—
13C2×C4×Q8×C11C2211yescharacteristic
14C2×C4×D4×C11C2211yescharacteristic
15C2×#67000—21no—
16C22×#4324C2211yescharacteristic
17C22×#4323C2212yes—
18#10623116—24no—
19#10623116—24no—
20C2×#66947—22no—
21C2×#66946—21no—
22C2×#66928C2212yes—
23C2×#66927C2211yescharacteristic
24C2×C4×D44C2212yes—
25C2×C4×Dic22C2211yescharacteristic
26C11×#396—22no—
27C11×#390C2212yes—
28C11×#389C2212yes—
29C11×#384C2211yescharacteristic
30C11×#380—21no—
31C2×#66896C2211yescharacteristic
32C4×#4324—24no—
33#10622628—24no—
34C2×#66874C2211yescharacteristic
35C2×#66874C2212yes—
36C2×#66873C2212yes—
37C2×#66873C2212yes—
38C2×#66873C2212yes—
39C2×#66872C2211yescharacteristic
40C2×C4⋊(C11⋊C8)C2211yescharacteristic
41C2×C4×C11⋊C8C2211yescharacteristic
42#10622167C2212yes—
43#10622162C2212yes—
44#10622162C2212yes—
45C22×Q8×C11D411yescharacteristic
46C22×D4×C11C2311yescharacteristic
47C23×C4×C11—21no—
48C22×D44—21no—
49C22×D44—21no—
50C22×D44C2312yes—
51C22×C4×D11—22no—
52C22×Dic22C2311yescharacteristic
53C11×C22⋊Q8—44no—
54C4×Q8×C11—22no—
55C4×Q8×C11—22no—
56C4×D4×C11—22no—
57C4×D4×C11—22no—
58C2×C11×C4⋊C4—21no—
59C2×C11×C4⋊C4—22no—
60C2×C11×C4⋊C4—21no—
61C2×C11×C4⋊C4D411yescharacteristic
62C2×C11×C4⋊C4D412yes—
63C2×C11×C4⋊C4C2311yescharacteristic
64C2×C11×C4⋊C4C2312yes—
65C2×C11×C22⋊C4—21no—
66C2×C11×C22⋊C4—22no—
67C2×C11×C22⋊C4—22no—
68C2×C42×C11C2312yes—
69C2×C42×C11C2311yescharacteristic
70#67000—28no—
71C2×#4324—22no—
72C2×#4324—22no—
73C2×#4324—22no—
74C2×#4323—24no—
75C2×#4323—24no—
76C2×#4323—42no—
77C2×(C2×D11)⋊C4—22no—
78C2×(C2×D11)⋊C4—22no—
79C2×(C2×D11)⋊C4—22no—
80C2×C4⋊Dic11C2311yescharacteristic
81C2×Dic11⋊C4—21no—
82C2×C4×Dic11—21no—
83C22×C11⋊C8C2311yescharacteristic
84C22×C11⋊C8C2312yes—
85C22×C11⋊C8C2311yescharacteristic
86#66947—216no—
87#66946—28no—
88#66928—24no—
89#66928—24no—
90#66927—22no—
91#66927D414yes—
92C4×D44—24no—
93C4×D44—24no—
94C4×Dic22—22no—
95C4×Dic22D414yes—
96C11×#95—22no—
97#66896—22no—
98#66874—22no—
99#66874—42no—
100#66873—24no—
101#66873—24no—
102#66873—42no—
103#66872—22no—
104C4⋊(C11⋊C8)—22no—
105C4×C11⋊C8—22no—
106#24186—111nomaximal, Sylow
107C24×C11—21no—
108C23×D11—22no—
109C2×Q8×C11—22no—
110C2×Q8×C11—22no—
111C2×Q8×C11—22no—
112C2×D4×C11—21no—
113C2×D4×C11—21no—
114C2×D4×C11—21no—
115C2×D4×C11—21no—
116C2×D4×C11—21no—
117C2×D4×C11#4112yes—
118C22×C4×C11—21no—
119C22×C4×C11—22no—
120C22×C4×C11—42no—
121C22×C4×C11—42no—
122C22×C4×C11—42no—
123C22×C4×C11C2×D412yes—
124C22×C4×C11C2×D412yes—
125C22×C4×C11C2×D411yescharacteristic
126C22×C4×C11C2×D411yescharacteristic
127C22×C4×C11C2411yescharacteristic
128C22×Dic11—21no—
129C22×Dic11—21no—
130C2×D44—24no—
131C2×D44—24no—
132C2×D44—22no—
133C2×D44—22no—
134C2×D44—22no—
135C2×D44—22no—
136C2×D44—22no—
137C2×D44—42no—
138C2×D44—42no—
139C2×D44—81no—
140C2×D44—81no—
141C2×D44C2×D414yes—
142C2×C4×D11—28no—
143C2×C4×D11—82no—
144C2×C4×D11—82no—
145C2×Dic22—21no—
146C2×Dic22—21no—
147C2×Dic22—21no—
148C2×Dic22—21no—
149C2×Dic22—21no—
150C2×Dic22C2×D412yes—
151C11×C4⋊C4—22no—
152C11×C4⋊C4—24no—
153C11×C4⋊C4—21no—
154C11×C4⋊C4—21no—
155C11×C4⋊C4—22no—
156C11×C4⋊C4—22no—
157C11×C4⋊C4—44no—
158C11×C4⋊C4—42no—
159C11×C4⋊C4—81no—
160C11×C22⋊C4—44no—
161C11×C22⋊C4—42no—
162C11×C22⋊C4—82no—
163C42×C11—22no—
164C42×C11—22no—
165C42×C11C2×D412yes—
166C42×C11C2×D412yes—
167#4324—28no—
168#4323—48no—
169(C2×D11)⋊C4—28no—
170(C2×D11)⋊C4—28no—
171(C2×D11)⋊C4—28no—
172C4⋊Dic11—21no—
173C4⋊Dic11C2×D412yes—
174Dic11⋊C4—24no—
175C4×Dic11—24no—
176C2×C11⋊C8—21no—
177C2×C11⋊C8—21no—
178C2×C11⋊C8—22no—
179C2×C11⋊C8—21no—
180C2×C11⋊C8—22no—
181C2×C11⋊C8—21no—
182C2×C11⋊C8—22no—
183C2×C11⋊C8—21no—
184C2×C11⋊C8#4112yes—
185C2×#559—111no—
186C2×#459—111no—
187C2×C4×SD16—111no—
188#2538D1111yescharacteristic, complemented
189#2042—112no—
190#2012—112no—
191#1984—111no—
192#1957—111no—
193#1879—112no—
194#1864—112no—
195#1803—111no—
196C23×C11—22no—
197C23×C11—41no—
198C23×C11—41no—
199C23×C11—41no—
200C23×C11—41no—
201C23×C11—41no—
202C23×C11—41no—
203C23×C11C22×D411yescharacteristic
204C22×D11—24no—
205C22×D11—42no—
206C22×D11—42no—
207C22×D11—42no—
208C22×D11—42no—
209C22×D11—42no—
210C22×D11—42no—
211Q8×C11—44no—
212D4×C11—22no—
213D4×C11—22no—
214D4×C11—41no—
215D4×C11—41no—
216C2×C4×C11—21no—
217C2×C4×C11—21no—
218C2×C4×C11—22no—
219C2×C4×C11—22no—
220C2×C4×C11—22no—
221C2×C4×C11—22no—
222C2×C4×C11—22no—
223C2×C4×C11—22no—
224C2×C4×C11—41no—
225C2×C4×C11—42no—
226C2×C4×C11—41no—
227C2×C4×C11—42no—
228C2×C4×C11—41no—
229C2×C4×C11—42no—
230C2×C4×C11—84no—
231C2×C4×C11#12012yes—
232C2×C4×C11#12012yes—
233C2×C4×C11#13512yes—
234C2×C4×C11AGL(1,ℤ/8)12yes—
235C2×C4×C11C22×D411yescharacteristic
236C2×C4×C11C22×D411yescharacteristic
237C2×C4×C11C2×#4111yescharacteristic
238C2×C4×C11C2×#4111yescharacteristic
239C2×C4×C11H(5,𝔽2)11yescharacteristic
240C2×C4×C112−1+411yescharacteristic
241C2×Dic11—22no—
242C2×Dic11—22no—
243C2×Dic11—41no—
244C2×Dic11—41no—
245C2×Dic11—41no—
246C2×Dic11—41no—
247D44—24no—
248D44—24no—
249D44—42no—
250D44—42no—
251D44—48no—
252D44—84no—
253C4×D11—88no—
254Dic22—22no—
255Dic22—22no—
256Dic22—41no—
257Dic22—41no—
258C11⋊C8—22no—
259C11⋊C8—41no—
260C11⋊C8—42no—
261C22×SD16—111no—
262C22×D8—112no—
263#559—224no—
264#559—224no—
265C2×#127—111no—
266C2×#124—112no—
267C2×#124—221no—
268C2×#123—222no—
269C2×C22⋊Q8—21no—
270C2×C4×Q8—111no—
271C2×C4×Q8C2×D1111yescharacteristic
272C2×C4×D4—112no—
273C2×C4×D4—221no—
274C2×C4×D4C2×D1111yescharacteristic
275#459—224no—
276C4×SD16—224no—
277C2×#106—111no—
278C2×C4⋊C8—111no—
279C2×#103—111no—
280C2×#103—112no—
281C2×#102—111no—
282C2×#102—112no—
283C2×#102—112no—
284C2×#102—112no—
285C2×C4×C8—111no—
286#396—22no—
287#390C2×D1112yes—
288#389C2×D1112yes—
289#384C2×D1111yescharacteristic
290#380—21no—
291#340—112no—
292#336—112no—
293#336—112no—
294C22×C11—42no—
295C22×C11—42no—
296C22×C11—42no—
297C22×C11—81no—
298C22×C11—81no—
299C22×C11—81no—
300C22×C11—81no—
301C22×C11C2×#12011yescharacteristic
302C22×C11#54811yescharacteristic
303C22×C11#55011yescharacteristic
304C22×C11C2×#13511yescharacteristic
305C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C11#57611yescharacteristic
307C22×C11#57711yescharacteristic
308C2×D11—44no—
309C2×D11—44no—
310C2×D11—44no—
311C2×D11—82no—
312C2×D11—82no—
313C2×D11—82no—
314C2×D11—82no—
315C4×C11—22no—
316C4×C11—42no—
317C4×C11—42no—
318C4×C11—82no—
319C4×C11—81no—
320C4×C11#44914yes—
321C4×C11#52511yescharacteristic
322C4×C11#53711yescharacteristic
323C4×C11#54611yescharacteristic
324C4×C11#54711yescharacteristic
325Dic11—42no—
326Dic11—42no—
327C22×Q8—111no—
328C22×Q8C22⋊D1111yescharacteristic
329C22×D4—112no—
330C22×D4—221no—
331C22×D4—221no—
332C22×D4C22×D1111yescharacteristic
333C23×C4—21no—
334C23×C4—222no—
335C2×SD16—222no—
336C2×SD16—222no—
337C2×SD16—222no—
338C2×D8—224no—
339C2×D8—224no—
340C2×D8—442no—
341C22×C8—111no—
342C22×C8—112no—
343C22×C8—111no—
344#127—114no—
345#127—222no—
346#124—224no—
347#124—224no—
348#124—228no—
349#123—2216no—
350C22⋊Q8—44no—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×Q8—114no—
354C4×Q8—222no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—224no—
358C4×D4—224no—
359C4×D4—228no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—111no—
364C2×C4⋊C4—221no—
365C2×C4⋊C4C22⋊D1111yescharacteristic
366C2×C4⋊C4C22⋊D1112yes—
367C2×C4⋊C4C22×D1111yescharacteristic
368C2×C4⋊C4C22×D1112yes—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—222no—
373C2×C22⋊C4—222no—
374C2×C22⋊C4—222no—
375C2×C42—221no—
376C2×C42C22×D1112yes—
377C2×C42C22×D1111yescharacteristic
378#106—222no—
379C4⋊C8—222no—
380#103—222no—
381#103—442no—
382#102—224no—
383#102—224no—
384#102—222no—
385#102—442no—
386C4×C8—222no—
387#95—22no—
388C2×C11—82no—
389C2×C11#240711yescharacteristic
390C2×C11C2×#44911yescharacteristic
391C2×C11#301211yescharacteristic
392C2×C11#336111yescharacteristic
393C2×C11#337311yescharacteristic
394C2×C11#339411yescharacteristic
395C2×C11#339911yescharacteristic
396D11—84no—
397C24—21no—
398C24—222no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—112no—
403C2×Q8—221no—
404C2×Q8—221no—
405C2×Q8—221no—
406C2×Q8—221no—
407C2×Q8—221no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—114no—
414C2×D4—224no—
415C2×D4—224no—
416C2×D4—222no—
417C2×D4—222no—
418C2×D4—222no—
419C2×D4—222no—
420C2×D4—222no—
421C2×D4—442no—
422C2×D4—442no—
423C2×D4—881no—
424C2×D4—881no—
425C2×D4#434312yes—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—228no—
432C22×C4—221no—
433C22×C4—221no—
434C22×C4—882no—
435C22×C4—882no—
436C22×C4C2×C22⋊D1112yes—
437C22×C4C2×C22⋊D1112yes—
438C22×C4C2×C22⋊D1111yescharacteristic
439C22×C4C2×C22⋊D1111yescharacteristic
440C22×C4C23×D1111yescharacteristic
441SD16—228no—
442D8—448no—
443C2×C8—112no—
444C2×C8—221no—
445C2×C8—221no—
446C2×C8—221no—
447C2×C8—222no—
448C2×C8—222no—
449C2×C8—221no—
450C2×C8—221no—
451C2×C8—222no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4—112no—
462C4⋊C4—224no—
463C4⋊C4—221no—
464C22⋊C4—44no—
465C22⋊C4—42no—
466C22⋊C4—82no—
467C22⋊C4—228no—
468C22⋊C4—228no—
469C22⋊C4—228no—
470C42—22no—
471C42—22no—
472C42—224no—
473C42D4×D1112yes—
474C42D4×D1112yes—
475C11#2418611yescharacteristic, complemented, Sylow
476C23—22no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—224no—
484C23—442no—
485C23—442no—
486C23—442no—
487C23—442no—
488C23—442no—
489C23—442no—
490C23C22×C22⋊D1111yescharacteristic
491Q8—44no—
492Q8—222no—
493Q8—222no—
494Q8—441no—
495Q8—441no—
496D4—22no—
497D4—22no—
498D4—41no—
499D4—41no—
500D4—224no—
501D4—224no—
502D4—442no—
503D4—442no—
504D4—448no—
505D4—884no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—84no—
521C2×C4—222no—
522C2×C4—222no—
523C2×C4—441no—
524C2×C4—441no—
525C2×C4—441no—
526C2×C4—441no—
527C2×C4—888no—
528C2×C4#6698512yes—
529C2×C4#6699012yes—
530C2×C4#6699012yes—
531C2×C4#6700312yes—
532C2×C4C2×D4×D1111yescharacteristic
533C2×C4C2×D4×D1111yescharacteristic
534C2×C4C2×#434311yescharacteristic
535C2×C4#6704011yescharacteristic
536C2×C4#41×D1111yescharacteristic
537C2×C4#6704211yescharacteristic
538C8—222no—
539C8—441no—
540C8—442no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—444no—
549C22—444no—
550C22—444no—
551C22—882no—
552C22—882no—
553C22—882no—
554C22—882no—
555C22C2×#6698511yescharacteristic
556C22C2×#6699011yescharacteristic
557C22#1062334811yescharacteristic
558C22C2×#6700311yescharacteristic
559C22#1062336511yescharacteristic
560C22#1062336611yescharacteristic
561C22#1062336811yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—81no—
566C4—82no—
567C4—442no—
568C4—442no—
569C4#1062265014yes—
570C4#1062315811yescharacteristic
571C4#1062316011yescharacteristic
572C4#1062316111yescharacteristic
573C4#1062316211yescharacteristic
574C2—82no—
575C2—884no—
576C2#1307196311yescharacteristic
577C2C2×#1062265011yescharacteristic
578C2#1307791711yescharacteristic
579C2#1308127611yescharacteristic
580C2#1308140811yescharacteristic
581C2#1308161211yescharacteristic
582C2#1308165811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#2538C2
socleC23×C11C22×D4
prime radicalC2×#10622628C2
hypercenter#2538D11
hypocenterC11#24186
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2538

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#10623116C2semidirect
3. C2×C4×#4324C2semidirect
4. #13069857C2semidirect
5. #13069692C2semidirect
6. #13069553C2semidirect
7. #13069421C2semidirect
8. #13068967C2semidirect
9. #13068876C2semidirect
10. #13068618C2semidirect
11. C11×#2538C2semidirect
188. #2538D11semidirect
475. C11#24186semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#106231162
C2×#106226282
C2×C4×#43242
#130698572
#130696922
#130695532
#130694212
#130689672
#130688762
#130686182
C11×#25382
#2418611